Shaykh al-'Alwān says in his explanation of the hadīth of Rasūlullah ﷺ : "...fast upon sighting it..." :
Some of the imāms like Ahmad and other than him used the hadīth as evidence that the sighting of the new moon in one land is sufficient to not have to sight it in other lands, and that is due to the generality of his (صلى الله عليه وسلم) saying: "fast upon sighting it."
...and ahl al 'ilm differed regarding this issue into [a number of] views:
The first is what has preceded and it is the view of Ahmad and the well-known view of the Māliki's, that if the people of a land see it (the new moon) then what becomes incumbent upon them (i.e. fasting or breaking fast) becomes incumbent upon [all] the other lands.
The second is that each land has its own sighting, and this is the view of ibn 'Abbās, al-Qāsim bin Muhammad, and Ishāq bin Rāhawayh, and the dalīl for this is what Muslim narrated in his sahīh from the path of Muhammad bin Abi Harmalah from Kurayb that Umm al Fadhl bint al-Hārith sent him to Mu'āwiyah in al-Shām. He (Kurayb) said: "So I arrived in Shām and completed her errand, and the new moon of Ramadān appeared while I was in Shām, and I saw the new moon on the night of Friday, then I arrived back in al-Madīnah at the end of the month, so 'Abdullah ibn 'Abbās (رضي الله عنه) questioned me, then mentioned the new moon, saying: 'when did you see the new moon?' I said:'we saw it on the night of friday', he said: 'You saw it yourself?', I said: 'yes and the people saw it and they fasted and Mu'āwiyah fasted', whereupon he [ibn 'Abbās] said: 'but we saw it on the night of Saturday so we'll continue fasting until we complete thirty days or we see it (the new moon of Shawwāl)'. I said: 'Are you not content with the sighting of Mu'āwiyah and his fasting?', so he said: 'No...this is how the Messenger of Allah ﷺ ordered us.' "
So ibn 'Abbās did not act upon the sighting of the people of al-Shām, while the ruler of the Muslims was one, and it wasn't reported from any of the sahābah that they opposed ibn 'Abbās [in this]. at-Tirmidhi stated in his Jāmi': "...and this hadīth is acted upon according to the people of knowledge, that the people of each land have their own sighting."
...and his saying "this is how the Messenger of Allah ﷺ ordered us" carries the possibility that ibn 'Abbās memorized words from the Messenger of Allah ﷺ that the people of a land aren't obligated to act upon the sighting of other lands, and it [also] carries the possibility that he said it from his own ijtihād and understanding.
The third view is that if the moon rises at the same point [as another land in which the new moon was sighted] then they must fast, and if not then no, and this is the well-known view of the Shāfi'i fuqahā and ibn Taymiyyah favored it. There are other views also, and the one most likely to be correct is the second view, and especially in this era when there became a ruler in each land ruling over it and many rely on calculation instead of sighting.
As for those who live in other than the lands of Islām then they should fast according to the closest Islāmic land, and if there isn't one then they follow the people of Makkah, and if they were to exert themselves in trying to sight the new moon and fast based on their surveying, it would be sufficient for them, and Allah knows best.
- Sharh Bulūgh-ul-Marām p. 797